Thursday, 3 September 2026

Find $\int (3x-5)^{12} dx$.

Suppose a 64 lb weight stretches a spring 6 inches in equilibrium and a dashpot provides a damping force of \(c\) lb for each ft/sec of velocity. (a) Write the equation of motion of the object and determine the value of \(c\) for which the motion is critically damped. (b) Find the displacement \(y\) for \(t > 0\) if the motion is critically damped and the initial conditions are \(y(0) = 1\) and \(y'(0) = 20\). (c) Find the displacement \(y\) for \(t > 0\) if the motion is critically damped and the initial conditions are \(y(0) = 1\) and \(y'(0) = -20\).

\(f(x) = x^3.\)

Let $D$ be the region in $xyz$-space defined by $1 \leq x \leq 2$, $0 \leq xy \leq 2$, and $0 \leq z \leq 1$. Evaluate $\iiint_D (x^2 y + 3xyz) dx dy dz$ by using the transformation $u = x$, $v = xy$, and $w = 3z$.

Determine whether \( \mathbf{F}(x, y) = \langle \sin x \cos y, \cos x \sin y \rangle \) is conservative.

Solve \(\frac{dy}{dx} = 3x^2y^2,\) given \(y(0) = \frac{1}{2}.\)

\(\int \frac{x^3}{\sqrt{1 + x^2}} dx.\)

Find $\int_e^{e^2} \frac{1}{x \ln x} dx$.

Find $D_x (\csc \sqrt{x})$.

Simplify: (a) $(5 + 3i)(2 – 7i)$, (b) $(4-3i)^2$, (c) $(1 + 2i)^3$.